One of my favourite soundtracks is "Doctor Who: Series 5", i.e. the first series with the 11th Doctor. That's all orchestral, but the KLF did a fun little song ("Doctorin' the TARDIS") with some basic lyrics: "Doctor Who-oo, hey, Doctor Who!" Chameleon Circuit are a band with a similar concept, and some of their lyrics are very clever. For instance, "Count the Shadows" is based on the episode Silence in the Library (where the 10th Doctor met River Song). This includes the lines:
"An invisible carnivorous aggressive armada,
And they go by the name of the Vashta Nerada."
They've released 2 albums so far, and I think my favourite song is "Mr Pond" (sung from Rory's point of view). Also, I can no longer hear the word 'boogie' without thinking of the song "Journey's End"!
In a different genre, someone recommended "Real Balls" by Doc Brown:
Initially, I wasn't impressed; I'm not a fan of "gangsta rap", and this seemed to be the same kind of posturing that I despise. However, stick with it: after about the 1 minute mark, it becomes clear that this song is a bit different. It's also a good example of a music video that helps to tell a story, rather than just dancing.
More generally, I've been listening to some comedy songs. I've seen Mitch Benn perform on stage a couple of times (at his own event and as part of The Now Show), and I've bought one of his albums: "Too Late to Cancel". There are some funny songs on there, although my favourite is probably the one that someone else sang ("Now He's Gone").
One of the webcomics I follow is The Oatmeal. The creator also collaborated on the Pterodactyl song:
I won't type the full title here, but suffice it to say that you probably shouldn't play it out loud at work.
If you liked that song, I also recommend Rachel Bloom. She's done a variety of songs, some of which I like more than others, but this is probably a good place to start:
It's a parody of a song that I haven't heard, but you don't need to be familiar with the original to understand this one. Again, the video adds to the song, which is always good to see.